Problem 21 - Essential Calculus Early Transcendentals

Question

Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$\sum_{n=1}^{\infty} \arctan n$$

Step-by-Step Solution

Answer

The series diverges since \( \lim_{n \to \infty} \arctan n = \frac{\pi}{2} \neq 0 \).

Step 1: Understand the Series Type

The series is given by \( \sum_{n=1}^{\infty} \arctan n \). This is an infinite series where the terms are the arctangent functions of natural numbers. To determine if it converges or diverges, we need to analyze the behavior of its terms as \( n \) approaches infinity.

Step 2: Apply Divergence Test

The divergence test states that if the limit of the terms of a series \( \lim_{n \to \infty} a_n eq 0 \), the series \( \sum_{n=1}^{\infty} a_n \) diverges. For our series, the terms \( a_n = \arctan n \). As \( n \to \infty \), \( \arctan n \to \frac{\pi}{2} \), which is not zero.

Step 3: Conclude the Series Behavior

Since \( \lim_{n \to \infty} \arctan n = \frac{\pi}{2} eq 0 \), the divergence test confirms that the series \( \sum_{n=1}^{\infty} \arctan n \) does not converge. Therefore, the series diverges.

Key Concepts

Divergence Test

The Divergence Test is a useful tool when analyzing the convergence or divergence of infinite series. This test is straightforward and begins by evaluating the limit of the individual terms of a series as they approach infinity. The rule is simple: if the limit of the terms of the series, denoted as \( a_n \), is not equal to zero, then the series diverges. This implies that the sum of the series cannot settle to a single finite number.

In the case of the series \( \sum_{n=1}^{\infty} \arctan n \), each term is the arctangent of \( n \). As \( n \) tends toward infinity, the arctangent also approaches a finite value, specifically \( \frac{\pi}{2} \), which is clearly not zero. Therefore, according to the Divergence Test, this series diverges. It's worth noting that if the limit were zero, this test alone would not be sufficient to indicate convergence—it would only confirm divergence if the limit isn't zero.

Infinite Series

An Infinite Series may seem daunting at first, as it involves summing an endless sequence of terms. These series are expressed in the form \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) represents each term. The real challenge is to determine if such a summation results in a finite number (converges) or continues indefinitely without settling on any specific value (diverges).

The condition under which an infinite series converges involves all its terms adding up to a particular value. Divergence, on the other hand, means that the series keeps growing indefinitely or behaves erratically. An infinite series can often be complex, much like the series involving the arctangent function, where the behavior of its components is crucial in determining the series' overall behavior.

Arctangent Function

The Arctangent Function is a vital component in this particular infinite series. The function \( \arctan x \) is the inverse of the tangent function and is commonly used to find the angle whose tangent is \( x \). For this function, as \( x \) increases, \( \arctan x \) progressively approaches \( \frac{\pi}{2} \) but never actually reaches it, due to the asymptotic nature of the function.

In the series \( \sum_{n=1}^{\infty} \arctan n \), the terms are the arctangent of each natural number. As you consider larger and larger values of \( n \), the terms \( \arctan n \) approach the limit \( \frac{\pi}{2} \). This property of getting closer to \( \frac{\pi}{2} \) without hitting zero plays a crucial role in determining the divergence of the series, as shown by the Divergence Test. Understanding how \( \arctan n \) behaves helps us conclude that the series does not converge.